Uniformly Continuous Functions on the Real Line

In summary, a uniformly continuous function is a type of function that shows minimal change in output for small changes in input. The main difference between uniform continuity and continuity is that uniform continuity requires the function to have a constant rate of change, while continuity only requires a limit at a given point. To determine if a function is uniformly continuous, one can use the definition of uniform continuity which states that for every epsilon greater than 0, there exists a delta greater than 0 such that for all x and y in the domain of the function, if the distance between x and y is less than delta, then the distance between the corresponding outputs f(x) and f(y) is less than epsilon. Uniform continuity is important because it guarantees smooth and predictable
  • #1
Euge
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Let ##f : \mathbb{R} \to \mathbb{R}## be a uniformly continuous function. Show that, for some positive constants ##A## and ##B##, we have ##|f(x)| \le A + B|x|## for all ##x\in \mathbb{R}##.
 
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  • #2
[tex]\forall \epsilon>0 \ \exists \delta> 0 \ \ |f(x+\delta)-f(x)|<\epsilon[/tex]
with no reard to individual value of x. Summing up this in x interval of [0,x] x>0
[tex]|f(x)-f(0)|\leq \frac{\epsilon}{\delta}|x|+\epsilon[/tex]
we get same formula for [x,0] x<0. So for any x
[tex]|f(x)| \leq \frac{\epsilon}{\delta}|x|+|f(0)|+\epsilon \leq A+B|x|[/tex]
where [tex]|f(0)|+\epsilon \leq A , \frac{\epsilon}{\delta}\leq B[/tex]
[EDIT]
Thanks to the sugegstion in post #13, I would add a sentense to the second line :
with no reard to individual value of x. Let us fix ##\epsilon## and ##\delta## which satisfy the inequality. Summing up this in x interval of [0,x] x>0
 
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  • #3
Isn't this just the definition of a uniformly continuous function?
 
  • #4
bob012345 said:
Isn't this just the definition of a uniformly continuous function?

I think the definition of uniformly continuous is a local definition - for any ##\epsilon## there exists ##\delta## such that ##|f(y)-f(x)|<\epsilon## if ##|x-y|<\delta##. The problem statement here is a statement about the whole function, not just two points that are very close together.

You have to do at least some work to get from one to the other.
 
  • #5
Office_Shredder said:
I think the definition of uniformly continuous is a local definition
I thought just plain old continuous was the local definition, and uniformly continuous was the global definition.

Office_Shredder said:
The problem statement here is a statement about the whole function
Not really; it's a property that must be satisfied at all values of ##x##, but so is ordinary continuity. But the definition of the property, for any given value of ##x##, only involves that value of ##x##; it does not involve the whole set of values of ##x## all at once.
 
  • #6
bob012345 said:
Isn't this just the definition of a uniformly continuous function?
I don't think so, because a uniformly continuous function does not have to be linear. The problem seems to be asking you to show that any uniformly continuous function can be approximated by a linear function.
 
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  • #7
Here is a reference that defines continuous and uniformly continuous:

https://people.math.wisc.edu/~jwrobbin/521dir/cont.pdf

A quick summary: if we look at epsilons and deltas, continuous means epsilon can vary, and delta can vary with both epsilon and ##x##; uniformly continuous means that epsilon can vary, and delta can vary with epsilon but not with ##x##; and the problem statement in the OP requires that there is a single epsilon and delta that work for all values of ##x##.
 
  • #8
PeterDonis said:
I thought just plain old continuous was the local definition, and uniformly continuous was the global definition.Not really; it's a property that must be satisfied at all values of ##x##, but so is ordinary continuity. But the definition of the property, for any given value of ##x##, only involves that value of ##x##; it does not involve the whole set of values of ##x## all at once.
Once you pick a choice of ##\epsilon## uniform continuity only says things about points that are close together. The fact that the same Delta and epsilon work is global, but the definition still says if two inputs are close together, so are the outputs.
 
  • #9
Office_Shredder said:
Once you pick a choice of ##\epsilon## uniform continuity only says things about points that are close together. The fact that the same Delta and epsilon work is global, but the definition still says if two inputs are close together, so are the outputs.
This is just as true of the problem statement in the OP. So I don't see that there is a useful "local" vs. "global" distinction here.
 
  • #10
No, the Op immediately gives a way to compare the output of two inputs no matter how far away they are. The definition only let's you compare the output of two inputs that are close together.

@anuttarasammyak's proof demonstrates this, they take something which is only true on a small interval, and they extrapolate it to demonstrate something that is true across any arbitrary interval. It's a bit subtle but it's there.
 
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  • #11
Office_Shredder said:
the Op immediately gives a way to compare the output of two inputs no matter how far away they are
The OP statement doesn't compare the function's values at two different values of ##x## at all.
 
  • #13
PeterDonis said:
I'm not sure this proof is valid.
It's conceptually a bit messy, IMO, as it leaves ##\epsilon## arbitrary. He should have let ##\epsilon =1##, for example.

Other than that, the idea is sound.
 
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  • #14
Euge said:
Let ##f : \mathbb{R} \to \mathbb{R}## be a uniformly continuous function. Show that, for some positive constants ##A## and ##B##, we have ##|f(x)| \le A + B|x|## for all ##x\in \mathbb{R}##.
What do you all think of ChatGPT's answer below?
Since ##f## is uniformly continuous, there exists a constant ##\delta > 0## such that, for any ##x,y \in \mathbb{R}## with ##|x - y| < \delta##, we have

$$|f(x) - f(y)| < 1.$$

Take any ##x \in \mathbb{R}##. Then, for any ##y \in \mathbb{R}## with ##|y| \le |x| + \delta##, we have

$$|f(x) - f(y)| < 1.$$

In particular, if we take ##y = 0##, we get

$$|f(x)| \le 1 + |f(0)|.$$

Therefore, we can take ##A = |f(0)|## and ##B = 1##. This shows that, for some positive constants ##A## and ##B##, we have

$$|f(x)| \le A + B|x|$$

for all ##x \in \mathbb{R}##.

Note that the choice of ##A## and ##B## is not unique: any constants ##A'## and ##B'## such that ##A' \ge A## and ##B' \ge B## will also work. For example, we can take ##A' = \max {|f(x)| : x \in \mathbb{R}}## and ##B' = \delta^{-1}##, where ##\delta## is the constant from the uniform continuity of ##f##.
 
  • #15
It's got the right shape, but serious errors.
 
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  • #16
PeterDonis said:
I don't think so, because a uniformly continuous function does not have to be linear. The problem seems to be asking you to show that any uniformly continuous function can be approximated by a linear function.
Bounded derivative is sufficient, per the mvt ( though function does not have to be differentiable to be UC.
 

Related to Uniformly Continuous Functions on the Real Line

1. What is a uniformly continuous function on the real line?

A uniformly continuous function on the real line is a function that maintains a constant rate of change over its entire domain. This means that the distance between any two points on the graph of the function will not change significantly, no matter how close together those points are.

2. How is uniform continuity different from regular continuity?

Uniform continuity is a stronger form of continuity than regular continuity. While both types of continuity require that the function's limit exists at a given point, uniform continuity also requires that the distance between the function's output values remains small even as the input values get closer together.

3. What is the importance of uniform continuity in mathematics?

Uniform continuity is important in mathematics because it allows us to make precise statements about the behavior of functions over their entire domain. It is also a key concept in the study of limits, derivatives, and integrals.

4. How can you determine if a function is uniformly continuous on the real line?

A function is uniformly continuous on the real line if and only if its derivative is bounded on the entire domain. This means that the rate of change of the function cannot become too large, ensuring that the function does not exhibit any sudden or drastic changes in behavior.

5. Can a function be uniformly continuous on a subset of the real line but not on the entire real line?

Yes, it is possible for a function to be uniformly continuous on a subset of the real line but not on the entire real line. This occurs when the function's derivative is bounded on the subset, but not on the entire domain. In this case, the function may exhibit sudden changes in behavior outside of the subset.

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